Question

Can somebody show me how to work this out? Find all of the zeros f(x) = 48x^4 - 52x^3 + 13x - 3

Answer 1

Dear UnknownRandom:
Do you happen to know and understand how to use synthetic division?

Answer 2

Yes

Answer 3

(52x(3)+13x)−3
=52x(3)+13x+(−3)
=169x+(−3)
=169x−3

Answer 4

Please give a medal and fan if this was helpful! :D

Answer 5

Look at the first and last coefficients of the given polynomial: 48 and -3. Multiply those together and then look for pairs of factors. -3(48)=-144. What are factors of -144?

This method depends somewhat on trial and error, but after you've identified some factors of -144 you could try them, one by one, as divisors in synth. div.

Nikkeyboo: I'd think medals and "fanning" would mean more if given spontaneously and not requested.

Answer 6

Easiest way is to use synthetic division and put 1 or -1 on the shelf.

Answer 7

I think so too. But what do you mean by "put 1 or -1 on the shelf?"

Answer 8

@mathmale That makes a lot of since. @Mertsj Alright I'll try that.

Answer 9

.5 | 48 -52 0 13 -3
24 -14 -7 3
48 -28 -14 6

Answer 10

Actually, it has to be 1/2 or -1/2. Sorry about that

Answer 11

So if 1/2 is a 0 then 2x-1 is a factor.

Answer 12

Nice work, Mertsj! Please explain 2 things to UR: how you arrived at that .5, and at that -.5, and how you knew from your example of synth. div. that .5 is a root. I'm interested in knowing your reasoning.

Answer 13

Note, PP and Mertsj, that 1/2 has numerator and denominator; that 1 in the numerator is a factor of 3, whereas that 2 in the denominator is a factor of 48 (but not of -3). Mertsj: Can you build on this explanation, please?

Answer 14

The possible rational roots are the factors of the constant term which is 3 divided by the factors of the coefficient of the leading term which is 48. I tried 1 and -1 and because of the way the numbers worked out, (and lots of experience) I tried 1/2. It worked. So I tried -1/2 and it worked as well. Then I have a quadratic trinomial which can be factored by the usual methods.

Answer 15

UR: I'd like to hear YOUR perspectives also! After all, it's your math problem.

Answer 16

I see what @Mertsj is doing. I was wondering if their was a faster way to do it.

Answer 17

Yes, Mertsj. To paraphrase what you've said: "Possible rational roots of the given polynomial have numerators which are factors of the last (the constant) term, and denominators which are factors of the first (the leading) term."

Answer 18

Factors of 3: 1 -1 3 -3
Factors of 48: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 16, -16, 24, -24, 48, -48,
3/6=1/2
-3/6=-1/2

Answer 19

Everybody wants a faster way to do stuff. But in the long run learning how to do problems correctly and well is more important than speed any day.

Answer 20

Mertsj: C O O L. Thank you.

Answer 21

@mathmale I agree 100%. Thanks again @mathmale and @Mertsj

Answer 22

yw

Answer 23

Bye!